Planar Barnette-type conjecture for sink-regular bipartite digraphs

Let $D=(V,A)$ be a sink-regular $(3,4)$-bipartite digraph that is planar, and let $Q_1,Q_2,Q_3$ be disjoint bases of $M_1(D,\1)$. A rounded $1$-factor $J$ is an object with associated set $\operatorname{dc}(J)$.

Barnette-type conjecture. There exists a rounded $1$-factor $J$ such that

$$\operatorname{dc}(J)=Q_1$$

and $D\setminus J$ is connected.

If true, this conjecture implies that every planar digraph whose dicuts have size at least three has three disjoint dijoins. Its resemblance to Barnette's conjecture for planar cubic bipartite graphs is noted in the source.